Remote Sensing of Water and Environment
Chapter 6: Microwave Dielectric Properties of Natural Earth Materials
Ardeshir Ebtehaj
University of Minnesota
1 Introduction
In this chapter we will learn about models of ϵ for different natural materials. Natural materials can be classified into three catagories:
- Homogeneous substances (pure water and ice)
- Electrolytic solution (salt or suger in water)
- Heterogeneous materials (sea ice, snow, soil, vegetation)
For a medium consisting of a host material with
, which is contained a homogeneous random concentration of ellipsoidal particles called inclusions with dielectric constant
. The dielectric constanct of the mixture is
where
is th effective or spatial average value of the dielectric constant of the medium and
is the fluctuating component.
is only a function of the shape and the oriantation of inclusions that might affect polarization and
. - If the orientation of the inclusion particles is random,
is independent of the direction and polarizaiton of the incident field. - The propagation constant of the medium γ is governed by
and volume scattering of the medium is a function of
.
In this chapter, we focus on attenuation as a result of
and not scattering.
For a plane wave in a lossy medium, we have
where
and
[Np/m]
[rad/m]
Ignoring scattering loss in the medium, one can have the following reprsentation of the power density throughout the depth of the medium
where
is the power absorption coefficient
.A related quantity of interest is penetration depth
. For a scatter-free medium:
[m]Note that
, where
is the skin depth. -----------------------------------------------------
Example 5-1: For a wet snow with density
, as the snow wetness chnages from 1 to 10%, the snow dielectric constant at
GHz chnages from
to
. Compute the penetration depth and explain the difference. Solution:
eps_1 = 1.7-1j*0.035; % 1% wetness
eps_10 = 2.8-1j*0.92; % 10% wetness
npp_1 =-imag(sqrt(eps_1));
delta_p_1 = 1/(2*alpha_1)
npp_10 =-imag(sqrt(eps_10));
delta_p_10 = 1/(2*alpha_10)
% When snow becomes wet, penetration depth decreases.
-----------------------------------------------------
2 Pure-Water Single-Debye Dielectric Model (
GHz)
For pure water the signle-Debye dielectric model (SD2M) is as follows:
: dielectric constant @ 
: dielectric constant @ 
: relaxation time constant [s]f: frequency [Hz]
where 
where T is in [
]
The term
is often called relaxation frequency that is
GHz
GHzTaking derivative of
with respect to f shows that
is maximum at
. Microwave spectra of the relative permittivity and loss factor of pure water at two different temperatures.
3 Saline-Water Double-Debye Dielectric Model (D3M,
GHz)
The unit psu is an acronym for practical salinity unit that is equivalent to grams of salt to kg of water. The averaged salinity in the global ocean is 35.5 psu, varying from less than 15 psu at the mouth of the rivers to more than 40 psu in the Dead Sea.
We have two relaxation terms
. Therefore,
shows two extrems at: The parameter functions are:
where:
The value of coefficients of D3M equation (Double-Debye dielectric model) is as follows.
Microwave spectra of (left) the permittivities and (right) loss factors of pure water (
) and seawater (
) at
C. Sea salinity is 32.54 psu.
T = 20; % temperature in C
f = 5; % frequency in GHz
S = 35; % salinity in parts per thousand
sig_35 = 2.903602 + 8.607e-2*T + 4.738817e-4*T^2 + -2.991e-6*T^3 + 4.3041e-9*T^4;
P = S*((37.5109 + 5.45216*S + 0.014409*S^2)/(1004.75 + 182.283*S + S^2));
alpha0 = (6.9431 + 3.2841*S + -0.099486*S^2) / (84.85 + 69.024*S + S^2);
alpha1 = 49.843 + -0.2276*S + 0.00198*S^2;
Q = 1 + ((alpha0*(T-15))/(T+alpha1));
a=[0.46606917e-2 -0.26087876e-4 -0.63926782e-5 0.63000075e1 0.26242021e-2 -0.42984155e-2 ...
0.34414691e-4 0.17667420e-3 -0.20491560e-6 0.58366888e3 0.12634992e3 0.69227972e-4 ...
0.38957681e-6 0.30742330e3 0.12634992e3 0.37245044e1 0.92609781e-2 -0.26093754e-1];
eps_w0 = 87.85306*exp(-0.00456992*T - a(1)*S - a(2)*S^2 - a(3)*S*T);
eps_w1 = a(4)*exp(-a(5)*T-a(6)*S-a(7)*S*T);
tau1 = (a(8)+a(9)*S)*exp(a(10)/(T+a(11)));
tau2 = (a(12)+a(13)*S)*exp(a(14)/(T+a(15)));
epsInf = a(16) + a(17)*T + a(18)*S;
%Complex Permitivity Calculation
eps = epsInf + ((eps_w0-eps_w1)./(1-1j*2*pi.*f.*tau1)) + ((eps_w1-epsInf)./(1-1j*2*pi.*f.*tau2)) + 1j*((17.9751*sigma)./f);
epsr = real(eps) % real part of relative dielectric constant
epsi = imag(eps) % imaginary part of relative dielectric constant
4. Dielectric Constant of Pure Ice
Unlike liquid water, whose relaxation frequencies lie in the microwave region, the relaxation frequency of pure ice
occurs in the kilohertz region. Thus, the pervious equation in MW band simplify as follows:
since
the single the Debye model expressions simplifies to: where
.
It was shown that
is independent of frequency in MW bands and the following model represents its dependence on the temperature.
where T is in
. Practically, we can ignore the dependancy to temperature and use
across the entire MW spectrum. There are more accurate model for
that accounts for infrared absorption spectrum as follows. where:
f is in GHZ.
Loss factor of pure ice.
Note: Fresh water ice often contains ionic impurities, such as dissolved salts, which causes its
to increase significantly in comparison with that for pure ice. Observational evidence suggests that
can be 2-8 times longer than that of pure ice.
%% Relative Dielectric Constant of Pure Ice
% T: Temperature in degree C
% epsr: real part of relative dielectric constant
% epsi: imaginary part of relative dielectric constant
%------------------------------------------------------------
T = T + 273; % represent temperature in Kelvin
alpha = (0.00504 + 0.0062*theta)*exp(-22.1*theta);
betaM = (B1/T) * exp(b/T) / (exp(b/T)-1)^2 + B2.*f.^2;
delBeta = exp(-9.963 + 0.0372*(T-273.16));
% epsr = 3.1884 + 9.1e-4*(T-273) .*f./f;
epsr = 3.1884 + 9.1e-4 *(T-273)
epsi = alpha./f + beta.*f
5 Dielectric Mixing Models for Heterogeneous Materials
A mixed material is defined by the host and inclusion materials.
- Host material: substance with the highest volume fraction
- Inclusion material: other existing materials
Dilectric constant of a mixture depends on:
- Dilectric constanct of individual substances
- Volume fractions of inclusions
- Spatial distribution of the inclusions
- Orientation of the inclusions relative to the direction of the incident electric-field
We need to find an average representation of dilectric constant based on the dilectric values of the host and inclusions.
Note: Here we assume that the dimensions of the inclusions are much smaller than the wavelength of the EM propagation field, thereby allowing to ignore volume scattering.
5-1 Randomly Oriented Ellipsoidal Inclusions
A host material containing only one type of inclusion is called a two-phased mixture consider that inclusions are ellipsoidal particles randomly dispersed within the host material. Each ellipsoid has dimension 2a, 2b, 2c along its major axes and host and inclusion material, have isotropic dilectric constants
, respectively. The inclusion concentration in a unit volume is defined by inclusion volume fraction
as follows: Thus fraction of the host material is
, where N is the number of ellipsoids per the volume. Randomly oriented ellipsoidal inclusions of identical size, shape, with dielectric constant
, dispersed in a host material with dielectric constant
. The ellipsoidal particles have semiaxes a, b, and c.
5.2 Tinga-Voss-Blossey (TVB) Formulas
The general dilectric formulation pertains to randomly dispersed confocal ellipsoids consisting of inner ellipsoid with dilectric constant
as shown in the following figure. Two-phase confocal ellipsoidal TVBmodel (Tinga et al., 1973) in which each ellipsoidal inclusion is surrounded by a shell of host material whose thickness is governed by the volume fraction of the host material.
- Inclusions are circular disk:
- Inclusions are needle shape:
Note: As is evident in all cases as
then
and when
then 
It is important to note that there are numerous models of dilectric constant of a mixture and all of them are in the following form:
where
linear model
refractive model
cubic model.
6 Dilectric constant of sea ice
Little progress has been made in chracterization of the dielectirc constant of sea ice due to its complexity.
- Sea ice is a hetrogeneous mixture of ice, liquid brine inclusion and air pockets.
Hetrogeneity of the inclusion varies in space and time significantly as the ice evolves
- Younge ice:
cm. - First year ice:
cm. - Multiyear ice:
cm.
The first two categories are similar in tems of the ice structure and dielectric properties. However, the brine concentration is different between multi-year sea ice and other two catagories, which is one of the main differences between them in terms their dilectric constant.
- The salinity profile of first-year ice typically decreases from about 5-16 psu near the surface to about 4-5 psu in the bulk of the ice and then increases rapidly to about 30 psu near the ice-water interface.
- In contrast, the salinity of multiyear ice usually is less than 1 psu in the surface layer and about 2-3 psu in the bulk portion. These values are representative of sea ice in the Arctic Ocean, where the salinity of liquid water is of the order of 32 psu.
In general
is a function of the following parameters:
of pure ice
of bine pockets- shape and the orientation of the brine inclusions (
) - fraction volume of brine (
) and spatial distribution
It is important to know that
is a function of tempreture. The reason is that because the temprerature drop causes more pockets of brine inclusions to freeze up and so the volume fraction of brine changes.
Temperature variation of the permittivity Temperature variation of the loss factor
for three types of sea ice at 10 GHz. of three types of sea ice at 10 GHz.
Measured permittivity and loss factor of simulated sea ice at 4.7 GHz, plotted as a function of brine volume fraction
Summary from the available observational data:
- For pure ice
at
and is independent of frequency. But loss factor changes as a function of frequency and temperature and is typically very small
. - For first year ice at
we can assume
from 1-20 GHz. The imaginary part can change from 0.1 to 0.25 at 10 GHz and
. - For multiyear sea ice
,
GHz -- where
.
7 Dilectric Constant of Snow
We treat dry and wet snow separatly. The reason is that permittivity of ice (
) and water (
) are drasically different. Thus a small amount of liquid water can change the dielectric constant of snow significantly.
7-1 Dry snow
Permittivity of dry snow
The dielectric constant of dry snow
depends on: - The dielectric constant of air
. - The dilectric constant of ice
and ice volume fraction
.
Assuring
(air) and grains of snow are circular,
model results in: For pure ice, we have
and thus we have,
,which is independent of temperature and frequency and only depends on the density.
Measured permittivity of dry snow as a function of snow density over frequency range 0.8--37 GHz.
An equally good fit to the data is provided (Matzler, 2006) as follows:
Loss factor of dry snow
When, ice is not pure, the loss factor
is non-zero and depends on both temperature and frequency. Using TBV models and assuming: Considering
, we have: Note that dry snow is a low-loss medium if
. Dry snow becomes more lossy as its density increases.
% The code computes the real and imaginary parts of the relative dielectric constant of Dry Snow
% Ps: Dry Snow Density in g/cm^3
% f: frequency in GHz (0.8 --37 GHz)
%epsr: real part of relative dielectric constant
%epsi: imaginary part of relative dielectric constant
[epsr_ice, epsi_ice] = RelDielConst_PureIce(T,f);
epsr_ds = 1 + 1.4667 .* vi + 1.435 .* vi.^3; % 0<vi<0.45
epsr_ds = (1+ 0.4759 .*vi).^3; % vi>0.45
epsi_ds = 0.34 * vi * epsi_ice ./(1- 0.42 * vi).^2;
7-2 Wet snow
In proximity of
, snow can hold water in liquid form
, where
is liquid water content of snow (volume fraction of water in snow). An extensive investigation of the dielectric behavior of natural snow was conducted by Hallikainen et al. (1983, 1984, 1986) using a free-space transmission technique. The permittivity and the dielectric loss factors of wet snow,
and
, were measured for 110 snow samples at nine frequencies between 4 and 18 GHz. Of these, 62 samples were also measured at 3 GHz and 37 GHz. Thus, although their results cover the range between 3 and 37 GHz, they are heavily influenced by the 4-18 GHz measurements. The range of
covered by the samples extended between 1 and 12 percent. Based on these measurments, a Debye-like model has been proposed for computation of wet snow dielectric constant as follows.
is relaxation frequency
[GHz]
[
]
[%]
Real and imaginary parts of the relative dielectric constant of snow as a function of liquid-water content at 6 GHz.
Spectral variation of the permittivity of wet snow with snow wetness as a parameter. The plots are based on the modified Debye-like model.
%%% Dilectric constant of wet snow.
% rho_s: Snow Density (g/cm^3)
% mv: volumetric water content (0<mv<30 in percentage scale)
% epsr: real part of relative permitivity
% epsi: imaginary part of relative permitivity
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
A1 = 0.78 + 0.03 .*f - 0.58e-3 .* f.^2;
A2 = 0.97 - 0.39e-2 .*f + 0.39e-3 .* f.^2;
B1 = 0.31 - 0.05 .*f + 0.87e-3 .* f.^2;
A = A1 .*(1.0 + 1.83*rho_s + 0.02*mv^1.015) + B1;
epsr = A + (B .* mv^x) ./ (1+(f/f0).^2)
epsi = (C .* (f/f0) .* mv^x) ./ (1+(f/f0).^2)
8 Dielectric Constant of Soil
8-1 Dry Soil
In the absence of liquid water, the microwave dielectric constant of soil,
, is essentially independent of both temperature and frequency. Moreover,
and
. Based on experimental measurements of several soil types, Dobson et al. (1985) determined that
can be modeled as where
is the soil bulk density, where
is the mass of the dry soil and
is the total volume. 8-2 Wet Soil
A wet-soil medium is a mixture of soil particles, air pockets, and liquid water.
The water contained in the soil usually is divided into two fractions: (a) bound water, and (b) free water.
Bound water refers to water molecules that are contained in the first few molecular layers surrounding the soil particles and therefore are tightly held by the soil particles due to the influence of matric and osmotic forces. Because the matric forces acting on a water molecule decrease rapidly with increasing distance from the soil particle, water molecules that are located several molecular layers away from soil particles are able to move within the soil medium with relative ease, and hence are referred to as “free” water.
The amount of water contained in the molecular layer adjoining the soil particles is directly proportional to the total surface area of the soil particles contained in a unit volume. The total surface area of the particles is, in turn, a function of the size distribution and mineralogy of the soil particles. For the most part, a soil is assigned to a textural class on the basis of its particle-size distribution.
Soil particles are classified as sand, silt, or clay, according to thier size. Because all soils contain a distribution of particle sizes, it is convenient to classify a soil by the weight-percent of the soil within each specific size category (sand, silt, or clay).
The two terms commonly used to characterize the moisture content of a soil sample are volumetric moisture
and gravimetric moisture
, Dielectric constant of wet soil dpends on
- f,T, salinity
- Total volumeric water content
- Relative fraction of bound and free water, which depends on the particle size distribution
- The bulk soil density
- Shape of the soil particles
Measured dielectric constant for five soils at 5 GHz (left) and measured dielectric constant as a function of volumetric moisture content for a loamy soil at four microwave frequencies (right) [Hallikainen et al., 1985].
Measured (left) permittivity and (right) loss factor of loamy soil as a function of frequency with volumetric moisture content as a parameter [Hallikainenet al., 1985].
Measured permittivity and loss factor of silt loam soil as a function of frequency with temperature as a parameter [from Hallikainen et al., 1984b].
Dobson et al. (1985) used a four-component model consisting of solid soil material, air, bound water, and free water.
where
is given by single Deby model as follows
,The expressions for
,
and
are given previously and
and where
and
are mass fraction of sand and clay.
Peplinski et al. (1995) conducted a study of several soil types, with specific focus on the dielectric behavior in the 0.3--1.3 GHz range abd suggested the following formula for the conductivity parameter:

%% dielectric constant of soil at a given temperature 0<T<40C,
% frequency, volumetric moisture content, soil bulk density, sand and clay fractions.
% rho_b: bulk density in g/cm3 (typical value is 1.7 g/cm3)
% S: Sand Fraction ( 0<S< 1)
% C: Clay Fraction ( 0<C< 1)
% mv: Volumetric Water content 0<mv<1
%epsr: real part of dielectric constant
%epsi: imaginary part of dielectric constant
f_hz = f * 1.0e9; % transform from GHz to Hz
beta1 = 1.27 - 0.519 * S - 0.152* C;
beta2 = 2.06 - 0.928 * S - 0.255 * C;
sigma_s = -1.645 + 1.939 * rho_b - 2.256*S + 1.594 * C; % Dobson et al. (1985)
sigma_s = 0.0467 + 0.22 * rho_b - 0.411*S + 0.661 *C; % Peplinski et al. (1995)
%Dielectric Constant of Pure Water
ew_0 = 88.045 - 0.4147 * T + 6.295e-4 * T^2 + 1.075e-5 * T^3;
tau_w = (1.1109e-10 - 3.824e-12*T +6.938e-14*T^2 - 5.096e-16*T^3)/2/pi;
epsrW = ew_inf +(ew_0-ew_inf)./(1 + (2*pi*f_hz*tau_w).^2);
epsiW = 2*pi*tau_w .*f_hz *(ew_0-ew_inf) ./(1 + (2*pi*f_hz*tau_w).^2) + ...
(2.65-rho_b)/2.65/mv * sigma_s ./(2*pi*eps_0*f_hz);
% calculating dielectric constant of soil
epsr = (1+ 0.66*rho_b + mv^beta1 * epsrW.^alpha - mv).^(1/alpha)

Comparison of values of
and
measured at 0.3 and 1.3 GHz with calculations based on Model 1 by Dobson (1985) and Model 2 by Peplinski et al. (1995). 9 Dielectric Constant of Vegetation
From the standpoint of wave propagation, a vegetation canopy is a dielectric mixture consisting of discrete dielectric inclusions (such as leaves, stalks, and fruit) distributed in a host material (air). In most canopies, the sizes of the inclusions are either comparable to or larger than the wavelength in the microwave region, which means that the canopy is an inhomogeneous, anisotropic medium. Propagation through such a medium entails both absorption and scattering.
9-1 Dielectric Constant of Canopy Constituents
Dielectric measurements of several grain types have been reported over a wide range of frequency band to 12.2 GHz in the microwave region (Nelson and Stetson, 1976).
Measured dielectric constant of red winter wheat heads as a function of moisture content [from Nelson and Stetson, 1976].
The moisture content of a vegetation material, such as leaves or stalks, is measured on a gravimetric wet weight basis (
) or on a volume basis (
). The two quantities are related by where
is the dry density of the solid material. A typical value of
for leaves is 0.3
. The range of
extends from 0 to about 0.9, and the corresponding range of
is from 0 to 0.7 (for
≈ 0.3). Using a waveguide transmission technique, Ulaby and Jedlicka (1984) measured the dielectric properties of leaves and stalks of corn and wheat over the 1.1 to 8.4 GHz range. Among the important parameters governing the behavior of
and
is the salinity S of the fluid samples extracted from the vegetation material. Measured moisture dependence of the dielectric constant of corn leaves at 1.5, 5.0, 8.0 GHz [Ulaby and Jedlicka, 1984].
Family of dielectric spectra for corn leaves at T = 22 ◦C and various moisture contents [El-Rayes and Ulaby, 1987].
Variation of er of corn leaves with decreasing temperature from 22 down to −32 C [El-Rayes and Ulaby, 1987].
9-2 Dielectric Model
Ulaby and El-Rayes (1987) introduced a linear model for the dielectric constant of vegetation in the form of an additive mixture of three components:
where
is a nondispersive residual component to be determined empirically,
and
are the complex dielectric constants of free water and bound water, respectively, and
and
are their associated volume fractions.
Free Water
For
C, the relaxation frequency of water is 18 GHz and thus Bound Water
Empirical Fits
By fitting the measured dielectric data to the dielectric model, the remaining quantities were found to have the following form
% Code computes the real and imaginary parts of the relative dielectric constant of vegetation material, such as corn leaves, in the microwave region.
%mg: Gravimetric moisture content 0< mg< 1
%eps_v_r: real part of dielectric constant
%eps_v_i: imaginary part of dielectric constant
sigma_i = 0.17.*S - 0.0013 .* S.^2;
eps_w_r = 4.9 + 74.4 ./( 1 + (f/18).^2);
eps_w_i = 74.4 .*(f/18) ./( 1 + (f/18).^2) + 18*sigma_i ./f ;
eps_b_r = 2.9 + 55*(1+ sqrt(f/0.36))./( (1+ sqrt(f/0.36)).^2 + (f/0.36));
eps_b_i = 55*sqrt(f/0.36) ./ ( (1+ sqrt(f/0.36)).^2 + (f/0.36));
v_fw = mg .*( 0.55 * mg - 0.076);
v_bw = 4.64 .*mg.^2 ./(1 + 7.36 * mg.^2);
eps_r = 1.7 - 0.74 *mg + 6.16 .* mg.^2;
eps_v_r = eps_r + v_fw .* eps_w_r + v_bw .*eps_b_r
eps_v_i = v_fw .* eps_w_i + v_bw .*eps_b_i